Stickers to Spell Word

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class Solution:
    def minStickers(self, stickers: List[str], target: str) -> int:
        m = len(stickers)
        mp = [[0] * 26 for _ in range(m)]
        dp = {"": 0}

        # count characters a-z for each sticker
        for i in range(m):
            for c in stickers[i]:
                mp[i][ord(c) - ord('a')] += 1

        return self.helper(dp, mp, target)

    def helper(self, dp, mp, target):
        if target in dp:
            return dp[target]

        ans = float('inf')
        n = len(mp)
        tar = [0] * 26

        for c in target:
            tar[ord(c) - ord('a')] += 1

        # try every sticker
        for i in range(n):
            # optimization
            if mp[i][ord(target[0]) - ord('a')] == 0:
                continue
            s = ""
            # apply a sticker on every character a-z
            for j in range(26):
                if tar[j] - mp[i][j] > 0:
                    s += (tar[j] - mp[i][j]) * chr(ord('a') + j)

            tmp = self.helper(dp, mp, s)
            if tmp != -1:
                ans = min(ans, 1 + tmp)

        dp[target] = -1 if ans == float('inf') else ans
        return dp[target]

10 Prerequisite LeetCode Problems

This involves dynamic programming and bitwise operations. Here are 10 problems to prepare for this:

  1. 72. Edit Distance: This problem is a classic dynamic programming problem that can help you understand the basic principles of DP.

  2. 322. Coin Change: This problem also involves finding the minimum number of items (coins) to sum to a target, which is similar to the sticker problem.

  3. 139. Word Break: This problem requires you to find if you can construct a target word from a list of words, which is related to the sticker problem.

  4. 140. Word Break II: This problem requires you to list all possible ways to construct the target word from a list of words. It’s a harder version of Word Break.

  5. 474. Ones and Zeroes: This problem involves a multi-dimensional knapsack problem, similar to the sticker problem.

  6. 983. Minimum Cost For Tickets: This problem requires dynamic programming to find the minimum cost to travel given various ticket options.

  7. 1349. Maximum Students Taking Exam: This problem involves bitwise manipulation and dynamic programming, which is similar to the sticker problem.

  8. 1879. Minimum XOR Sum of Two Arrays: This problem also involves bitwise operations and dynamic programming.

  9. 518. Coin Change 2: This problem is a classic dynamic programming problem which asks the number of combinations that make up a certain amount.

  10. 1049. Last Stone Weight II: This problem is about partitioning a set of numbers into two subsets to minimize the difference between the sums of these two subsets.

Problem Analysis and Key Insights

What are the key insights from analyzing the problem statement?

Problem Boundary

What is the scope of this problem?

How to establish the boundary of this problem?

Problem Classification

Problem Statement:We are given n different types of stickers. Each sticker has a lowercase English word on it.

You would like to spell out the given string target by cutting individual letters from your collection of stickers and rearranging them. You can use each sticker more than once if you want, and you have infinite quantities of each sticker.

Return the minimum number of stickers that you need to spell out target. If the task is impossible, return -1.

Note: In all test cases, all words were chosen randomly from the 1000 most common US English words, and target was chosen as a concatenation of two random words.

Example 1:

Input: stickers = [“with”,“example”,“science”], target = “thehat” Output: 3 Explanation: We can use 2 “with” stickers, and 1 “example” sticker. After cutting and rearrange the letters of those stickers, we can form the target “thehat”. Also, this is the minimum number of stickers necessary to form the target string. Example 2:

Input: stickers = [“notice”,“possible”], target = “basicbasic” Output: -1 Explanation: We cannot form the target “basicbasic” from cutting letters from the given stickers.

Constraints:

n == stickers.length 1 <= n <= 50 1 <= stickers[i].length <= 10 1 <= target.length <= 15 stickers[i] and target consist of lowercase English letters.

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Distilling the Problem to Its Core Elements

Can you identify the fundamental concept or principle this problem is based upon? Please explain. What is the simplest way you would describe this problem to someone unfamiliar with the subject? What is the core problem we are trying to solve? Can we simplify the problem statement? Can you break down the problem into its key components? What is the minimal set of operations we need to perform to solve this problem?

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?

Terminology

Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Provide names by categorizing these cases

What are the edge cases?

What are the key insights from analyzing the different cases?

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Simple Explanation

Can you explain this problem in simple terms or like you would explain to a non-technical person? Imagine you’re explaining this problem to someone without a background in programming. How would you describe it? If you had to explain this problem to a child or someone who doesn’t know anything about coding, how would you do it? In layman’s terms, how would you explain the concept of this problem? Could you provide a metaphor or everyday example to explain the idea of this problem?

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

Can you identify the key terms or concepts in this problem and explain how they inform your approach to solving it? Please list each keyword and how it guides you towards using a specific strategy or method.

How did you infer from the problem statement that this problem can be solved using ?

Simple Explanation of the Proof

I’m having trouble understanding the proof of this algorithm. Could you explain it in a way that’s easy to understand?

Stepwise Refinement

  1. Could you please provide a stepwise refinement of our approach to solving this problem?

  2. How can we take the high-level solution approach and distill it into more granular, actionable steps?

  3. Could you identify any parts of the problem that can be solved independently?

  4. Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Identify Invariant

What is the invariant in this problem?

Identify Loop Invariant

What is the loop invariant in this problem?

Thought Process

Can you explain the basic thought process and steps involved in solving this type of problem?

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

Establishing Preconditions and Postconditions

  1. Parameters:

    • What are the inputs to the method?
    • What types are these parameters?
    • What do these parameters represent in the context of the problem?

  2. Preconditions:

    • Before this method is called, what must be true about the state of the program or the values of the parameters?
    • Are there any constraints on the input parameters?
    • Is there a specific state that the program or some part of it must be in?

  3. Method Functionality:

    • What is this method expected to do?
    • How does it interact with the inputs and the current state of the program?

  4. Postconditions:

    • After the method has been called and has returned, what is now true about the state of the program or the values of the parameters?
    • What does the return value represent or indicate?
    • What side effects, if any, does the method have?

  5. Error Handling:

    • How does the method respond if the preconditions are not met?
    • Does it throw an exception, return a special value, or do something else?

Problem Decomposition

  1. Problem Understanding:

    • Can you explain the problem in your own words? What are the key components and requirements?

  2. Initial Breakdown:

    • Start by identifying the major parts or stages of the problem. How can you break the problem into several broad subproblems?

  3. Subproblem Refinement:

    • For each subproblem identified, ask yourself if it can be further broken down. What are the smaller tasks that need to be done to solve each subproblem?

  4. Task Identification:

    • Within these smaller tasks, are there any that are repeated or very similar? Could these be generalized into a single, reusable task?

  5. Task Abstraction:

    • For each task you’ve identified, is it abstracted enough to be clear and reusable, but still makes sense in the context of the problem?

  6. Method Naming:

    • Can you give each task a simple, descriptive name that makes its purpose clear?

  7. Subproblem Interactions:

    • How do these subproblems or tasks interact with each other? In what order do they need to be performed? Are there any dependencies?

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Code Explanation and Design Decisions

  1. Identify the initial parameters and explain their significance in the context of the problem statement or the solution domain.

  2. Discuss the primary loop or iteration over the input data. What does each iteration represent in terms of the problem you’re trying to solve? How does the iteration advance or contribute to the solution?

  3. If there are conditions or branches within the loop, what do these conditions signify? Explain the logical reasoning behind the branching in the context of the problem’s constraints or requirements.

  4. If there are updates or modifications to parameters within the loop, clarify why these changes are necessary. How do these modifications reflect changes in the state of the solution or the constraints of the problem?

  5. Describe any invariant that’s maintained throughout the code, and explain how it helps meet the problem’s constraints or objectives.

  6. Discuss the significance of the final output in relation to the problem statement or solution domain. What does it represent and how does it satisfy the problem’s requirements?

Remember, the focus here is not to explain what the code does on a syntactic level, but to communicate the intent and rationale behind the code in the context of the problem being solved.

Coding Constructs

Consider the following piece of complex software code.

  1. What are the high-level problem-solving strategies or techniques being used by this code?

  2. If you had to explain the purpose of this code to a non-programmer, what would you say?

  3. Can you identify the logical elements or constructs used in this code, independent of any programming language?

  4. Could you describe the algorithmic approach used by this code in plain English?

  5. What are the key steps or operations this code is performing on the input data, and why?

  6. Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

  1. Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.

  2. Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.

  3. Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

  1. Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.

  2. In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.

  3. Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.

Q&A

Similar Problems

Can you suggest 10 problems from LeetCode that require similar problem-solving strategies or use similar underlying concepts as the problem we’ve just solved? These problems can be from any domain or topic, but they should involve similar steps or techniques in the solution process. Also, please briefly explain why you consider each of these problems to be related to our original problem.

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class Solution:
    def minStickers(self, stickers: List[str], target: str) -> int:
        tcount = Counter(target)
        def getScore(s):
            temp = dict(tcount)
            sdict = defaultdict(int)
            res = 0
            for c in s:
                if c in temp and temp[c] > 0:
                    temp[c] -= 1
                    res += 1
                    sdict[c] += 1
            return (res, sdict)
        stickers = [getScore(s) for s in stickers]
        stickers.sort(key = lambda x: x[0], reverse = True)
        stickers = [x[1] for x in stickers if x[0] > 0]
        opt = [stickers[0]]
        for i in range(1, len(stickers)):
            if opt[-1].keys() == stickers[i].keys() :
                continue
            opt.append(stickers[i])
        stickers = opt[:]
            
        seen = set()
        q = deque([target])
        step = 0
        while q:
            n = len(q)
            step += 1
            for i in range(n):
                cur = q.popleft()
                if cur in seen:
                    continue
                seen.add(cur)
                for stick in stickers:
                    if cur[0] not in stick:
                        continue
                    new = str(cur)
                    for s in stick:
                        new = new.replace(s,'', stick[s])
                    if new == "":
                        return step
                    if new not in seen:
                        q.append(new)
        return -1

Problem Classification

This problem is from the Computer Science domain and more specifically it belongs to the Data Structures and Algorithms category. It involves concepts of Strings, Combinatorial Search, and Optimization.

Here are the ‘What’ components of the problem:

  1. A list of n stickers is given. Each sticker has a lowercase English word on it. The length of each word is within the range from 1 to 10. You have an infinite number of each sticker.

  2. A target string is given. The length of the target is within the range from 1 to 15. The target string consists of lowercase English letters.

  3. The task is to form the target string by cutting individual letters from the stickers and rearranging them.

  4. Each sticker can be used more than once.

  5. The goal is to return the minimum number of stickers that are needed to form the target string. If it’s impossible to form the target string with the given stickers, return -1.

Based on these components, we can further classify this problem as a Combinatorial Search Problem because it involves exploring all possible combinations of stickers to form the target string. It’s also an Optimization Problem as we need to minimize the number of stickers used. Moreover, it can be seen as a String Manipulation Problem as it requires cutting and rearranging letters from the given strings (stickers) to form a new string (target). Lastly, given the problem constraints and the need to explore a large number of possibilities efficiently, this problem will likely involve Dynamic Programming to store and reuse solutions to subproblems.

Domain Classification: This problem falls into the domain of Combinatorial Optimization, which is a part of computer science and operations research. It is also related to the domain of Dynamic Programming, as we need to make optimal decisions at each step to achieve the overall optimal solution.

What components:

  1. Input: We are given n different types of stickers where each sticker has a lowercase English word on it. The quantity of each sticker is infinite.
  2. Output: We are to spell out the given target string by cutting and rearranging individual letters from the stickers. The output should be the minimum number of stickers that are needed to spell out the target string.
  3. Constraints: If the target string cannot be spelled out using the stickers, the function should return -1.
  4. Additional Information: The target string is a concatenation of two random words. All words and target are chosen randomly from the 1000 most common US English words.

Problem Classification: The problem can be classified as a Dynamic Programming problem as it involves making an optimal choice at each step (choosing a sticker) to ensure that the overall solution is optimal (minimum number of stickers used). It also demonstrates optimal substructure and overlapping subproblems, two key characteristics of Dynamic Programming problems. The problem can also be considered a Combinatorial Optimization problem as it seeks to minimize the number of stickers used, which is a type of optimization, and involves combinatorial elements (picking combinations of stickers).

  1. Optimization Problem: The aim is to find a solution that minimizes the number of stickers used.

  2. Combinatorial Problem: The problem requires generating and examining various combinations of stickers to meet the target.

  3. String Manipulation: The problem requires operations on strings (stickers), such as comparisons, replacements, and counting characters.

  4. Search Problem: The problem involves searching through the space of all combinations of stickers to find a solution that satisfies the condition of spelling out the target string.

  5. Graph Theory Problem: The problem can be seen as a Breadth-First Search on a graph, where each node represents a state (the current target string), and each edge represents an operation (applying a sticker).

These classifications overlap to some extent, as many complex problems can be seen as combinations of smaller problems from different domains.

Language Agnostic Coding Drills

  1. List, String and Dictionary Manipulation: Basic understanding of manipulating Python Lists, Strings, and Dictionaries is required. Concepts such as adding, removing, sorting elements, and iterating through lists, strings, and dictionaries are integral to this problem.

  2. Counter objects: The Counter class in Python’s collections module is used for counting hashable objects. Here it is used to count the occurrences of characters in the target string.

  3. DefaultDict objects: Another dictionary subclass is defaultdict, which calls a factory function to supply missing values. In the given problem, defaultdict is used with an integer, which defaults to 0.

  4. Lambda Functions: Lambda functions are small anonymous functions. They can take any number of arguments but can only have one expression. They are used here to sort the stickers based on their score.

  5. Queue Data Structure: Understanding of the Queue data structure, specifically the deque (double-ended queue) in Python’s collections module, is necessary. It’s used for the Breadth-First Search (BFS) algorithm.

  6. Breadth-First Search (BFS): This is a fundamental graph traversal algorithm that explores all the vertices of a graph in breadth-first order. It’s used here to find the minimum number of stickers that can form the target string.

  7. Greedy Algorithm: The approach of picking the best option at each decision point with the hope that these local decisions will lead to a global optimum. The algorithm here is a greedy one as it always chooses the sticker with the maximum overlap with the current string.

  8. Dynamic Programming (DP): The problem involves solving overlapping subproblems and uses memoization to store the results of these subproblems to avoid re-computation. Here, the DP concept is used to optimize the number of stickers needed to form a target string.

When approaching the problem:

  1. First, the frequency of each character in the target string is determined.

  2. For each sticker, it is scored by the sum of frequencies of its characters present in the target string. Only characters present in the target string are kept in the stickers.

  3. The stickers are then sorted by their scores in descending order. This is to ensure that stickers with higher overlap with the target string are used first.

  4. A breadth-first search (BFS) is then conducted where, at each level, all possible new strings formed by using one of the stickers are explored.

  5. Each new string is added to the queue, and this process continues until either the queue is empty (in which case return -1) or an empty string is found (return the current level in BFS).

  6. To avoid repeating work, strings that have been seen before are stored in a set and skipped in future iterations.

Targeted Drills in Python

  1. List, String and Dictionary Manipulation
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# Drill: Create a list, add elements, remove elements, sort the list and iterate through it
my_list = [3, 2, 1]
print(my_list)
my_list.append(4)
print(my_list)
my_list.remove(2)
print(my_list)
my_list.sort()
print(my_list)
for i in my_list:
    print(i)
  1. Counter Objects
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# Drill: Count the number of occurrences of each character in a string
from collections import Counter
str1 = 'abccba'
counter = Counter(str1)
print(counter)
  1. DefaultDict Objects
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# Drill: Create a defaultdict with integer default values and manipulate it
from collections import defaultdict
default_dict = defaultdict(int)
default_dict['a'] += 1
default_dict['b'] += 2
print(default_dict)
  1. Lambda Functions
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# Drill: Use lambda function to sort a list of tuples based on the second element
tuples_list = [(1, 'b'), (2, 'a'), (3, 'c')]
sorted_list = sorted(tuples_list, key=lambda x: x[1])
print(sorted_list)
  1. Queue Data Structure
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# Drill: Implement a queue using collections.deque and perform operations: append, popleft
from collections import deque
queue = deque()
queue.append('a')
queue.append('b')
print(queue.popleft())
print(queue)
  1. Breadth-First Search (BFS)
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# Drill: Implement a simple BFS on a graph represented as an adjacency list
graph = {'A': ['B', 'C'], 'B': ['D', 'E'], 'C': [], 'D': [], 'E': ['F'], 'F': []}
visited = []
queue = deque()
start_node = 'A'
queue.append(start_node)
while queue:
    node = queue.popleft()
    if node not in visited:
        visited.append(node)
        neighbours = graph[node]
        for neighbour in neighbours:
            queue.append(neighbour)
print(visited)
  1. Greedy Algorithm
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# Drill: Implement a simple greedy algorithm to choose maximum number of activities (Activity Selection Problem)
activities = [(1, 2), (3, 4), (0, 6), (5, 7), (8, 9), (5, 9)]
activities.sort(key = lambda x: x[1])
i = 0
print(activities[i])
for j in range(len(activities)):
    if activities[j][0] >= activities[i][1]:
        print(activities[j])
        i = j
  1. Dynamic Programming (DP)
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# Drill: Implement a simple DP solution for the Fibonacci sequence
def fib(n, dp):
    if n <= 1:
        return n
    if dp[n] != -1:
        return dp[n]
    dp[n] = fib(n - 1, dp) + fib(n - 2, dp)
    return dp[n]
n = 10
dp = [-1 for i in range(n + 1)]
print(fib(n, dp))

After practicing these drills, you should have a solid understanding of the concepts required to implement the final solution. You can then try to combine these concepts to implement the final solution for the problem.